**Degree, regular, degree sequence**

The *degree* of a vertex is the number of incident edges it has.

The *First theorem of graph theory* relates degrees to edges. It states that the sum of the degrees of the vertices is twice the number of edges.

If every vertex of a graph has the same degree, then that graph is called *regular.* In an *n*-regular graph, all vertices have degree *n.*

Thus, in a 2-regular graph, all vertices have degree 2. |

In a 4-regular graph all vertices have degree 4, and so forth. |

For example, the table beside the graph at the left lists the degree of each vertex.

When the numbers in the degree column are rearranged into non-increasing order, the list becomes: 4, 4, 3, 3, 2, 2, 1, 1, 1, 1. This is the degree sequence for the graph.

Copyright © 1999-2000 SciMathMN